The Poincaré Conjecture, Assembled

Everything is in place. This chapter states the theorem the guide set out to prove, and assembles its proof from the chapters that built it, one step at a time. Nothing new is proved here. The point is to see the whole argument at once: ten steps, each a theorem proved earlier, and the threads of the guide (heat, compactness, monotonicity, curvature, scaling, topology, variation, entropy, linearisation) meeting where the proof needs them.

By the end of this chapter you will be able to:

• state the Poincaré conjecture and say why the smooth version proves it;
• give the proof in ten steps, naming for each the theorems that deliver it and the chapters where they are proved;
• say where each hypothesis of the theorem is used, and where it is not;
• trace each thread of the guide to the step of the proof where it ends;
• say which parts of Perelman's work the Poincaré conjecture does not need.

3.1A proof as a graph

3.2The theorem

The proof works with smooth manifolds and produces a diffeomorphism. That is enough: in dimension three every topological manifold has a smooth structure, unique up to diffeomorphism (Moise, 1952; 7A.9 The Poincaré Conjecture, Precisely), so a closed simply connected topological 3-manifold is homeomorphic to a smooth one, which the proof shows is diffeomorphic to $S^3$. Throughout, $M$ is closed, connected, orientable (every simply connected manifold is) and smooth.

3.3The proof in ten steps

Figure 3.1 draws the ten steps as a pipeline, and Figure 3.2 the dependencies between the chapters that prove them.

3.4Where the hypotheses are used

The theorem has two hypotheses beyond being a 3-manifold: $M$ is closed and simply connected.

• Closed is used throughout: the maximum principle at a spatial maximum (11A.4 Maximum Principles under Ricci Flow), the $\mathcal W$-entropy as an integral over $M$ with a lower bound for $\mu$ on the initial metric (12A.4 κ-Noncollapsing), finite volume and the count of surgeries (12B.5 Ricci Flow with Surgery for All Time). Flows on noncompact manifolds need different hypotheses.
• Simply connected is used only at the end, in two places: to make $\pi_3(M)$ nonzero, so that the width is nontrivial and finite extinction applies (step 9), and to discard every summand except $S^3$ (step 10). Steps 1–8 hold for every closed orientable 3-manifold. The flow with surgery does not know whether $M$ is simply connected; only the reading of its record does.
• Dimension three is used in steps 3, 6, 7 and 8: Hamilton–Ivey pinching, the classification of $\kappa$-solutions and of three-dimensional shrinkers, and necks of the form $S^2\times I$ whose central spheres are the spheres of the prime decomposition.

3.5Where the threads ended

The guide was built from threads, each carried from a first appearance to a destination in the proof. Here is where each one arrived.

thread	where it ends	the step
H heat and diffusion	the conjugate heat equation, Perelman's Harnack inequality (12A.2 Ricci Flow as a Gradient Flow, 12A.6 Pseudolocality)	5, 8
C compactness	canonical neighbourhoods (12B.3 The Canonical Neighbourhood Theorem)	6, 7
M monotonicity	$\mathcal F$, $\mathcal W$ and reduced volume (12A.3 The 𝓦-Entropy–12A.5 Reduced Distance and Reduced Volume)	5, 8
Q curvature as a quadratic form	pinching and the soliton equation (11A.5 Hamilton–Ivey Pinching, 11B.1 Ricci Solitons)	3, 6
S scaling and blow-up	rescaling at singularities, surgery scales (11B.4 Singularities, 12B.5 Ricci Flow with Surgery for All Time)	4, 8
T topology	reading off the topology (12C.1 Reading Off the Topology)	10
V variational structure	width and min–max (12C.2 Finite Extinction)	9
E entropy	$\mathcal W$ and $\mu$ (12A.3 The 𝓦-Entropy)	5
L linearisation	short-time existence (11A.3 Short-Time Existence and Uniqueness)	2

The contradiction–compactness template [CC] of 2A.5 Quantifiers and the Shape of a Proof appears at full strength in step 7.

3.6What the Poincaré conjecture does not need

Perelman's work proves more than the Poincaré conjecture, and some of it is not needed here. The long-time analysis of the flow (II, §§6–8), the thick–thin decomposition and the collapsing theorem belong to geometrization (12C.4 Geometrization); for a simply connected manifold the flow becomes extinct, and there is no long time to analyse. Perelman pointed this out himself in his third preprint, where he also noted that for a homotopy sphere one can avoid the Kneser finiteness theorem, and that with an extinction time already bounded, the time-dependent parameters of 12B.5 Ricci Flow with Surgery for All Time can be replaced by single values. Pseudolocality (12A.6 Pseudolocality) enters only through the standard solution, and the later classification of $\kappa$-solutions by Brendle and others (12B.2 The Structure of κ-Solutions) is not needed at all.

3.7History

Perelman posted his three preprints between November 2002 and July 2003. Between 2003 and 2006, Bruce Kleiner and John Lott, John Morgan and Gang Tian, and Huai-Dong Cao and Xi-Ping Zhu wrote detailed accounts, which together established that the proof was complete. Perelman was awarded a Fields Medal at the International Congress of Mathematicians in Madrid in 2006, and the Clay Mathematics Institute announced its Millennium Prize for the proof in March 2010. He declined both (11B.5 Hamilton’s Program in 2002).

3.8Exercises

Solution

(a) 2. (b) 3 (and through it 6–7). (c) 5. (d) 6. (e) 7. (f) 8. (g) 9. (h) 10.

Solution

Steps 1–8 go through: they hold for every closed orientable 3-manifold. Step 9 fails: $T^3$ is aspherical, its sweepouts are trivial, and the flow need not become extinct (for a flat metric it is static). Step 10 would also fail, since $\pi_1(T^3) = \mathbb{Z}^3$. In the long run the flow with surgery on $T^3$ collapses, and the analysis of 12C.4 Geometrization recognises it as a graph manifold.

Solution

By Moise's theorem every topological 3-manifold has a smooth structure, unique up to diffeomorphism. So a closed simply connected topological 3-manifold is homeomorphic to a smooth one with the same properties, which is diffeomorphic, hence homeomorphic, to $S^3$. In dimension four and higher, topological manifolds need not have smooth structures, or may have several, so the reduction fails (in dimension four the topological Poincaré conjecture is Freedman's theorem, while the smooth one is open).

Solution

$S^2\times S^1$: the sphere factor shrinks, a neckpinch-like singularity forms everywhere at once, the manifold is covered by necks and is discarded at the singular time (12B.4 Surgery); every step up to 8 applies. Lens spaces with round metrics shrink to points, a $\kappa$-solution picture of step 6. The two facts: (step 9) $\pi_3(M) \cong \mathbb{Z} \neq 0$ by Hurewicz, so a nontrivial sweepout class exists; (step 10) $\pi_1(M) = 1$, so in a connected sum of spherical space forms and copies of $S^2\times S^1$ every summand is $S^3$.

Solution

One version. Put any metric on $M$ and let it evolve by the Ricci flow, which smooths curvature like heat (steps 1–3). Where curvature blows up, zoom in: Perelman's entropy shows the zoomed-in pictures never collapse, so they converge to a short list of models (steps 4–6). Every region of high curvature therefore looks like a thin neck or a cap, so you can cut along the necks, cap them off and keep going, and this happens only finitely often in any finite time (steps 7–8). Because $M$ is simply connected, the area of the smallest family of spheres sweeping it out shrinks at a fixed rate, so eventually nothing is left (step 9). Every piece that was cut off or vanished was a sphere-like piece, and gluing them back is a connected sum, so $M$ was a sphere all along (step 10).